Probability Generating Function Research Articles

The discrete-time Geom/G/1 queue with multiple adaptive vacations and server Setup/Closedown times

In this paper, we discuss the discrete-time Geom/G/1 queue with multiple adaptive vacations and server setup/closedown times. We consider two types of closedown time policies which differ from each other in the completion instant of the closedown times and contrast the effects of them on the queueing system. We derive the probability generating functions (PGFs) and stochastic decomposition results of the steady-state queue length and waiting time distributions through the method of embedded Markov chain in both cases. Many discussed models related to Geom/G/1 queue are special cases of our model.

Number of Complete N-ary Subtrees on Galton-Watson Family Trees

We associate with a Bienayme-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer \(N\), define a complete \(N\)-ary tree to be the family tree of a deterministic branching process with offspring generating function \(s^N\). We study the random variables \(V_{N,n}\) and \(V_N\) counting the number of disjoint complete \(N\)-ary subtrees, rooted at the ancestor, and having height \(n\) and \(\infty\), respectively. Dekking (1991) and Pakes and Dekking (1991) find recursive relations for \(P(V_{N,n}>0)\) and \(P(V_N>0)\) involving the offspring probability generation function (pgf) and its derivatives. We extend their results determining the probability distributions of \(V_{N,n}\) and \(V_N\). It turns out that they can be expressed in terms of the offspring pgf, its derivatives, and the above probabilities. We show how the general results simplify in case of fractional linear, geometric, Poisson, and one-or-many offspring laws.

On the joint distribution of runs in the sequence of Markov-dependent multi-state trials

Under different popular counting schemes of runs we have obtained probability-generating function (pgf) of the joint distribution of runs in a sequence of Markov-dependent multi-state trials. A method of conditional pgf's is used to obtain these joint distributions. Expanding the matrix polynomial of degree n , involved in the pgf, a recurrent formula for evaluation of exact probability distribution is obtained. To show the feasibility and compatibility of the results obtained, the joint distributions are numerically evaluated.

Gestational mutations and carcinogenesis

We present a mathematical formulation to evaluate the effects of gestational mutations on cancer risk. The hazard or incidence function of cancer is expressed in terms of the Probability Generating Function (PGF) of the number of normal and mutated cells at birth. Using Filtered Poisson Process Theory, we obtain the PGF for several models for the accumulation of gestational mutations. In particular, we develop expressions for the hazard function when one or two successive mutations could occur during gestation. We also calculate the hazard when the background gestational mutation rates are increased due to exposure to mutagens, such as prenatal radiation. To illustrate the use of our models, we apply them to colorectal cancer in the SEER database. We find that the proportion of cancer risk attributable to developmental mutations depends on age and that it could be quite significant when gestational mutation rates are high. The analysis of the SEER data also shows that gestational mutations could contribute to inter-individual variations in colorectal cancer risk.

Some Illustrative Classroom Examples Regarding Sums of Discrete Random Variables With Finite Support

In many probability and mathematical statistics courses the probability generating function (PGF) is typically overlooked in favor of the more utilized moment generating function. However, for certain types of random variables, the PGF may be more appealing. For example, sums of independent, non-negative, integer-valued random variables with finite support are easily studied via the PGF. In particular, the exact distribution of the sum can easily be calculated. Several illustrative classroom examples, with varying degrees of difficulty, are presented. All of the examples have been implemented using the R statistical software package.

The null distribution of Kendall's rank correlation statistic in the presence of ties

We present new techniques for computing exact p-values of Kendall’s rank correlation statistic when ties are present. An explicit formula for the probability-generating function of this statistic in the case of ties in one ranking is derived. This allows for computation of exact p-values for sample sizes up to 100, regardless of the tie structure. When ties are present in both rankings, one has to fall back on enumeration methods. We discuss how a large part of this enumeration can be avoided. The exact results for ties in one ranking are used to develop an efficient Monte Carlo simulation algorithm for approximating p-values in case of ties in both rankings. Finally, we shortly discuss computer implementations of our methods, which are available on the internet.

An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions

For a broad class of discrete- and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some multiserver queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the probability generating functions (PGFs) of the stationary numbers in queue and in system, as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with batch Markovian arrival process (BMAP) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.

Modes of Reproduction and the Accumulation of Deleterious Mutations With Multiplicative Fitness Effects

Abstract Mutational load depends not only on the number and nature of mutations but also on the reproductive mode. Traditionally, only a few specific reproductive modes are considered in the search of explanations for the maintenance of sex. There are, however, many alternatives. Including these may give radically different conclusions. The theory on deterministic deleterious mutations states that in large populations segregation and recombination may lead to a lower load of deleterious mutations, provided that there are synergistic interactions. Empirical research suggests that effects of deleterious mutations are often multiplicative. Such situations have largely been ignored in the literature, since recombination and segregation have no effect on mutation load in the absence of epistasis. However, this is true only when clonal reproduction and sexual reproduction with equal male and female ploidy are considered. We consider several alternative reproductive modes that are all known to occur in insects: arrhenotoky, paternal genome elimination, apomictic thelytoky, and automictic thelytoky with different cytological mechanisms to restore diploidy. We give a method that is based on probability-generating functions, which provides analytical and numerical results on the distributions of deleterious mutations. Using this, we show that segregation and recombination do make a difference. Furthermore, we prove that a modified form of Haldane’s principle holds more generally for thelytokous reproduction. We discuss the implications of our results for evolutionary transitions between different reproductive modes in insects. Since the strength of Muller’s ratchet is reduced considerably for several forms of automictic thelytoky, many of our results are expected to be also valid for initially small populations.

Modes of reproduction and the accumulation of deleterious mutations with multiplicative fitness effects.

Mutational load depends not only on the number and nature of mutations but also on the reproductive mode. Traditionally, only a few specific reproductive modes are considered in the search of explanations for the maintenance of sex. There are, however, many alternatives. Including these may give radically different conclusions. The theory on deterministic deleterious mutations states that in large populations segregation and recombination may lead to a lower load of deleterious mutations, provided that there are synergistic interactions. Empirical research suggests that effects of deleterious mutations are often multiplicative. Such situations have largely been ignored in the literature, since recombination and segregation have no effect on mutation load in the absence of epistasis. However, this is true only when clonal reproduction and sexual reproduction with equal male and female ploidy are considered. We consider several alternative reproductive modes that are all known to occur in insects: arrhenotoky, paternal genome elimination, apomictic thelytoky, and automictic thelytoky with different cytological mechanisms to restore diploidy. We give a method that is based on probability-generating functions, which provides analytical and numerical results on the distributions of deleterious mutations. Using this, we show that segregation and recombination do make a difference. Furthermore, we prove that a modified form of Haldane's principle holds more generally for thelytokous reproduction. We discuss the implications of our results for evolutionary transitions between different reproductive modes in insects. Since the strength of Muller's ratchet is reduced considerably for several forms of automictic thelytoky, many of our results are expected to be also valid for initially small populations.

Random-allocation and urn models

We review some urn and random-allocation models, mostly using probability generating function (PGF) methods. We begin by formulating a basic problem which can be thought of as either an urn or a random-allocation model; a PGF solution to it is outlined. When the compartments in the latter model are no longer homogeneous, the multivariate PGF can still be derived, though the algebra becomes cumbersome. Some results are given for the case where there are two types of compartment and for the case where there are two types of ball. Some comments are offered on the Frobenius–Harper property of PGFs.

Random-allocation and urn models

We review some urn and random-allocation models, mostly using probability generating function (PGF) methods. We begin by formulating a basic problem which can be thought of as either an urn or a random-allocation model; a PGF solution to it is outlined. When the compartments in the latter model are no longer homogeneous, the multivariate PGF can still be derived, though the algebra becomes cumbersome. Some results are given for the case where there are two types of compartment and for the case where there are two types of ball. Some comments are offered on the Frobenius–Harper property of PGFs.

Performance Analysis of a Discrete-Time Two-Phase Queueing System

This paper introduces the modeling and analysis of a discrete-time, two-phase queueing system for both exhaustive batch service and gated batch service. Packets arrive at the system according to a Bernoulli process and receive batch service in the first phase and individual services in the second phase. We derive the probability generating function (PGF) of the system size and show that it is decomposed into two PGFs, one of which is the PGF of the system size in the standard discrete-time Geo/G/1 queue without vacations. We also present the PGF of the sojourn time. Based on these PGFs, we present useful performance measures, such as the mean number of packets in the system and the mean sojourn time of a packet.

Hit statistics in FH-CDMA unslotted packet networks

This correspondence is concerned with the statistical dependence of hits due to multiple-access interference in the time-unslotted frequency-hopping (FH) packet networks. Relying on the technique of joint probability generating functions (PGFs), we give a formal proof that if the ratio of the number of interfering packets to the number of frequency slots K/q is held constant, hits due to multiple-access interference are asymptotically independent as q/spl rarr//spl infin/. We also derive expressions for the correlation coefficients of hits in the packet.

A performance analysis of a discrete-time queueing system with server interruption for modeling wireless ATM multiplexer

In this paper, we study the effect of channel errors on wireless ATM. The salient characteristic of the wireless channel is that it is time varying. This variability is modeled by a Markov chain with two states corresponding to high and low error states, respectively. The channel is modeled as being synchronous with the basic time unit being the slot. The transitions from high to low error states, and vice versa occur at the slot boundaries. It is assumed that no transmission over the channel is possible during the high-error state. The information source feeding into the channel is modeled as the superposition of independent sources. Each source is controlled by a two-state Markov chain which operates in synchronism with, but statistically independent of the channel. In the source Off state, no data is emitted while, in the On state, the source generates a random number of packets in a slot each of which fits into a channel slot. It is assumed that the traffic generated by all the sources will form a single global queue which will be served by the two-state Markov channel. A discrete-time queuing analysis derives the probability generating function (PGF) of the queue length under the assumption of an infinite buffer. From the PGF, we determine mean queue length and mean delay. Finally, numerical results are presented to demonstrate the effect of wireless channel error characteristics on performance.

A Priority Discrete Queueing Model for Multimedia Multiplexers

Multiplexers have been extensively modeled as discrete time queueing systems. In this article, we model a multimedia multiplexer handling traffic of two classes. One class represents real-time traffic, e.g., packets of live audio or video transmissions, and the other nonreal-time traffic, e.g., packets of file transfer transmissions. These packets arrive into the multiplexer in batches. In each time slot, one batch of each class arrive. The multiplexer gives service priority to class-1 packets over class-2. The demands of each class are in conflict with that of the other, and thus they are treated by the multiplexer differently. The multiplexer is thus modeled as a (preemptive) priority discrete queueing system with simultaneous batch arrivals and geometric service time. The system occupancy is analyzed and the joint probability generating function (PGF) of the number of packets of each class is derived. From this PGF, marginal PGFs of interest are obtained. The results for deterministic service time, most suitable for ATM purposes, are readily obtainable as a special case from the results of this article.

Cascade Theory of Substitution Effects in Nonequilibrium Polycondensation Systems

The substitution effect of neighboring groups on the statistical quantities in nonequilibrium condensation polymerization is analyzed by the vectorial cascade theory. In equilibrium condensation polymerization, the substitution is affected by the conversional distribution of building (monomer) units. In nonequilibrium condensation polymerization, however, it needs to be expressed by the distribution function of the local connection of links. The average degree of polymerization and the critical point of gelation are obtained by generalizing the conventional cascade theory. The molecular mass distribution is evaluated by direct recurring calculations with a cutoff in the original probability generation function or in the Groebner base. Compared with conventional treatment by Lagrange's expansion, the evaluation method developed here requires less computer memory. This allows us to complete the calculations within a reasonable time. As a typical example, formulation of the probability generating function for the first-shell substitution effect in trifunctional unit systems and the calculation of their molecular mass distribution are given.

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

The application of probability-generating functions to linear–quadratic radiation survival curves

Purpose : To illustrate how probability-generating functions (PGFs) can be employed to derive a simple probabilistic model for clonogenic survival after exposure to ionizing irradiation. Methods : Both repairable and irreparable radiation damage to DNA were assumed to occur by independent (Poisson) processes, at intensities proportional to the irradiation dose. Also, repairable damage was assumed to be either repaired or further (lethally) injured according to a third (Bernoulli) process, with the probability of lethal conversion being directly proportional to dose. Using the algebra of PGFs, these three processes were combined to yield a composite PGF that described the distribution of lethal DNA lesions in irradiated cells. Results : The composite PGF characterized a Poisson distribution with mean, alphaD + betaD 2, where D was dose and alpha and beta were radiobiological constants. This distribution yielded the conventional linear–quadratic survival equation. To test the composite model, the derived distribution was used to predict the frequencies of multiple chromosomal aberrations in irradiated human lymphocytes. The predictions agreed well with observation. This probabilistic model was consistent with single-hit mechanisms, but it was not consistent with binary misrepair mechanisms. Conclusions : A stochastic model for radiation survival has been constructed from elementary PGFs that exactly yields the linear–quadratic relationship. This approach can be used to investigate other simple probabilistic survival models.

Delay analysis of selective repeat ARQ for a Markovian source over a wireless channel

In this paper, we analyze the mean delay experienced by a Markovian source over a wireless channel with time-varying error characteristics. The wireless link implements the selective-repeat automatic repeat request (ARQ) scheme for retransmission of erroneous packets. We obtain good approximations of the total delay, which consists of transport and resequencing delays. The transport delay, in turn, consists of queueing and transmission delays. In contrast to previous studies, our analysis accommodates both the inherent correlations between packet interarrival times (i.e., traffic burstiness) and the time-varying nature of the channel error rate. The probability generating function (PGF) of the queue length under the "ideal" SR ARQ scheme is obtained and combined with the retransmission delay to obtain the mean transport delay. For the resequencing delay, the analysis is performed under the assumptions of heavy traffic and small window sizes (relative to the channel sojourn times). The inaccuracy due to these assumptions is observed to be negligible. We show that ignoring the autocorrelations in the arrival process or the time-varying nature of the channel state can lead to significant underestimation of the delay performance, particularly at high channel error rates. Some interesting effects of key system parameters on the delay performance are observed.